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Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Span basis swap!

PureU1.VectorLikeEvenPlane.span_basis_swap!

physicsquantum field theoryrelativitybasis linear

Project documentation

The basis formed out of our basisa vectors. -/ noncomputable def basisaAsBasis : Basis (Fin (succ n) ⊕ Fin n) ℚ (PureU1 (2 * succ n)).LinSols := basisOfLinearIndependentOfCardEqFinrank (@basisa_linear_independent n) basisa_card /-! ## F. Every Lienar solution is the sum of a point from each plane -/ lemma span_basis (S : (PureU1 (2 * n.succ)).LinSols) :...

Exact Lean statement

lemma span_basis_swap! {S : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
    (hS : ((FamilyPermutations (2 * n.succ)).linSolRep
    (Equiv.swap (evenShiftFst j) (evenShiftSnd j))) S = S') (g : Fin n.succ → ℚ) (f : Fin n → ℚ)
    (h : S.val = P g + P! f) : ∃ (g' : Fin n.succ → ℚ) (f' : Fin n → ℚ),
      S'.val = P g' + P! f' ∧ P! f' = P! f +
      (S.val (evenShiftSnd j) - S.val (evenShiftFst j)) • basis!AsCharges j ∧ g' = g

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
lemma span_basis_swap! {S : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)    (hS : ((FamilyPermutations (2 * n.succ)).linSolRep    (Equiv.swap (evenShiftFst j) (evenShiftSnd j))) S = S') (g : Fin n.succ → ℚ) (f : Fin n → ℚ)    (h : S.val = P g + P! f) : ∃ (g' : Fin n.succ → ℚ) (f' : Fin n → ℚ),      S'.val = P g' + P! f' ∧ P! f' = P! f +      (S.val (evenShiftSnd j) - S.val (evenShiftFst j)) • basis!AsCharges j ∧ g' = g := by  let X := P! f + (S.val (evenShiftSnd j) - S.val (evenShiftFst j)) • basis!AsCharges j  have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) := by    apply Submodule.add_mem    exact (P!_in_span f)    exact (smul_basis!AsCharges_in_span S j)  have hXsum := (Submodule.mem_span_range_iff_exists_fun ℚ).mp hX  obtain ⟨f', hf'⟩ := hXsum  use g  use f'  change P! f' = _ at hf'  erw [hf']  simp only [and_self, and_true, X]  rw [← add_assoc, ← h]  apply swap!_as_add at hS  exact hS
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/QFT/QED/AnomalyCancellation/Even/BasisLinear.lean:922-942

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